I have been searching for an analogue of chain rule for Sobolev functions. While in general, I could not find chain rule for composition for arbitrary functions (and neither do I expect such a result to hold), I found one result for the composition of a Lipschitz continuous function with a Sobolev function. I state it verbatim from here$^1$.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a Lipschitz continuous function. Then, for every $u \in W^{1, 1}_{loc} \left( \mathbb{R}^N, \mathbb{R}^d \right)$, the composite function $v = f \circ u$ is in $W^{1, 1}_{loc} \left( \mathbb{R}^N \right)$ and for a.e. $x \in \mathbb{R}^N$ the restriction of $f$ to the affine space $$T^u_x := \left\lbrace w \in \mathbb{R}^d | w = u (x) + \nabla u(x)z, \text{ for some } z \in \mathbb{R}^N \right\rbrace$$ is differentiable at $u(x)$ and $$\nabla (f \circ u ) (x) = \nabla_u (f|_{T^u_x}) (u(x)) \nabla u(x).$$
Here, $\nabla$ is the derivative of the function.
I cannot see how does the right hand side always make sense? If we look closely, $T^u_x$ need not be full-dimensional. How do we understand $\nabla_u(f|_{T^u_x})$? Moreover, how does one compose the two derivatives if they are not defined on appropriate dimensional spaces?
Any insights into this would be appreciated!
${}^1$: Leoni, Giovanni; Morini, Massimiliano, Necessary and sufficient conditions for the chain rule in $W^{1,1}_{loc}(\mathbb R^N;\mathbb R^d)$ and $BV_{loc}(\mathbb R^N;\mathbb R^d)$. J. Eur. Math. Soc. (JEMS) 9, No. 2, 219-252 (2007). Zbl 1135.26011, doi:10.4171/JEMS/78